12x12 Queens Puzzle #116
Difficulty: medium. Place 12 queens on this 12x12 board so that no two share a row, column or color region and none touch — including diagonally. This puzzle has exactly one solution, and it has been verified solvable by deduction alone: the complete 24-step reasoning is written out below.
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Step-by-step solution for puzzle #116
What follows is a complete solve of puzzle #116, worked the way a person works it rather than the way a computer does. It runs to 24 deductions — 12 queen placements and 12 elimination steps that narrow the board between them. Each step names the rule that forces it, so nothing below is a guess, and the hardest idea required is the proof by contradiction.
Techniques this board needs: last square in a region, region confined to one line, region shadow, pairing argument on rows, pairing argument on columns, line confined to one region, proof by contradiction.
Show the full walkthrough (15 steps) — contains the solution
Step 1. The board opens by cascading. Each of these placements is forced the moment the one before it is marked, so the whole run is really a single sweep repeated: indigo has one square left, so its queen takes row 1, column 1; mint has one square left, so its queen takes row 4, column 4; rose has one square left, so its queen takes row 12, column 11; teal has one square left, so its queen takes row 11, column 7. After each one, clear that row, that column, the rest of the colour and the ring of neighbours before looking for the next. The last of them seats the queen on row 11, column 7.
Step 2. Slate cannot leave column 2 — every one of its open squares (2 open squares) sits there. That column belongs to slate, so every square in it owned by another colour is dead: 5 squares in all.
Step 3. The khaki region has 7 squares left, spread across rows 5–9, columns 5, 6, 8, and 9. You do not need to know which of them takes the queen: every one of those options attacks the same 1 square, so they can be crossed out now.
Step 4. Wherever sea green ends up among its 2 squares, the same squares fall within reach of it — 3 squares in total, impossible regardless of how the colour resolves.
Step 5. Orange is down to 4 candidates in rows 8–10, columns 5 and 8, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about orange itself.
The board after step 5 — 4 of 12 queens placed. Step 6. Amber and blue are now boxed into rows 2 and 3 and nothing else. That is 2 colours needing 2 rows, so they fill those rows between them — which locks every remaining colour out and removes 5 squares at once.
Step 7. The slate region is down to one open square, at row 5, column 2. Every colour holds exactly one queen, so that square is forced. Cross out the whole of row 5 and column 2, the remaining slate squares, and the ring of neighbours around row 5, column 2.
Step 8. Khaki is down to 5 candidates in rows 6–8, columns 5, 6, 8, and 9, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about khaki itself.
Step 9. All of the pale yellow region's remaining squares (4 open squares) lie in column 3. Its queen therefore has to be somewhere in that column, which means no other colour may use it: 1 square come off.
Step 10. Blue and sea green are now boxed into columns 10 and 12 and nothing else. That is 2 colours needing 2 columns, so they fill those columns between them — which locks every remaining colour out and removes 6 squares at once.
The board after step 10 — 5 of 12 queens placed. Step 11. Red is down to 5 candidates in rows 6, 8, and 9, columns 8 and 9, and they overlap enough to cover 1 square elsewhere. Cross those out without deciding anything about red itself.
Step 12. Column 9 has only red squares left (3 open squares). The column needs a queen and only red can provide one, which pins that colour to this column and kills its squares elsewhere: 2 squares.
Step 13. Now a short proof by contradiction. Suppose the amber queen went to row 2, column 6. Follow the forced moves that would create and the red region is left with nowhere to put its queen — impossible. So that square is ruled out for good.
Step 14. Test row 3, column 6 for amber: place a queen there, run the forced consequences through, and the red region is left with nowhere to put its queen. The assumption breaks, so the square is eliminated.
Step 15. From here the board cascades. Keep alternating the region sweep and the line sweep and 7 queens fall out in order without any new idea being needed: amber has one square left, so its queen takes row 2, column 5; orange has one square left, so its queen takes row 9, column 8; red has one square left, so its queen takes row 6, column 9; khaki has one square left, so its queen takes row 8, column 6; sea green has one square left, so its queen takes row 10, column 12; blue has one square left, so its queen takes row 3, column 10; pale yellow has one square left, so its queen takes row 7, column 3. After each one, clear that row, that column, the rest of the colour and the ring of neighbours before looking for the next. The last of them seats the queen on row 7, column 3.
The board after step 15 — 12 of 12 queens placed.
That completes the board: 12 queens seated, one in every row, one in every column and one in every colour, with no two of them touching. Notice that nothing in the sequence above required a guess — each placement followed from an elimination already on the grid, which is what “single solution, solvable by logic” actually means in practice.
About 12x12 Queens puzzle #116
This is puzzle number 116 in the 12x12 collection, rated medium. The grid holds 144 squares split between twelve colored regions, and a finished board carries twelve queens — one in every row, one in every column, one in every color, and never two of them touching, diagonals included.
Puzzle #116 has been checked to admit exactly one arrangement of queens, and every step toward it is deducible from what is already on the board. There is no point at which you are expected to pick a square and hope. The only thing that differs between boards is how quickly the first certainty surfaces, and that depends entirely on the shapes the colors happen to take here.
How the colors divide this board
From smallest to largest, the regions on puzzle #116 are: indigo (1 square), mint (1 square), rose (1 square), slate (6 squares), teal (7 squares), blue (11 squares), khaki (12 squares), sea green (13 squares), orange (14 squares), red (25 squares), pale yellow (26 squares), amber (27 squares). An even split would hand every color 12 squares, so the indigo region is running 11 squares under that average while the amber region is the roomiest area on the grid with 27 squares.
One queen per color, regardless of acreage: that is what turns this uneven split into a solving order. Indigo is working with 1 square while amber has 27 to spare, a gap of 26. Attack the cramped end and the roomy regions tend to resolve themselves.
Square count is only half the story — how a color is arranged counts too. Indigo is the tidiest shape on #116, packing 100% of its 1×1 bounding box, whereas khaki straggles across its 5×6 box at just 40%. Tidy blocks are easy to work around; scattered ones touch more lines and resist resolution.
Region by region
Here is the full footprint of every color on puzzle #116, smallest first. Reading these spans before you place anything is the quickest way to see which parts of the grid are genuinely contested.
Indigo occupies 1 square within rows 1–1, columns 1–1. It never leaves row 1, so its queen is locked to that row.
Mint: 1 square across rows 4–4, columns 4–4. It never leaves row 4, so its queen is locked to that row.
Rose covers 1 square, spanning 1 row and 1 column (rows 12–12, columns 11–11). It never leaves row 12, so its queen is locked to that row.
Slate occupies 6 squares within rows 3–5, columns 1–2.
Teal: 7 squares across rows 11–12, columns 3–7.
Blue covers 11 squares, spanning 4 rows and 3 columns (rows 1–4, columns 10–12).
Khaki occupies 12 squares within rows 5–9, columns 4–9.
Sea green: 13 squares across rows 7–12, columns 9–12.
Orange covers 14 squares, spanning 5 rows and 6 columns (rows 8–12, columns 5–10).
Red occupies 25 squares within rows 2–9, columns 7–12.
Pale yellow: 26 squares across rows 6–12, columns 1–6.
Amber covers 27 squares, spanning 5 rows and 10 columns (rows 1–5, columns 1–10).
Where to start on this board
The most constrained color is where a solve should begin, and on #116 that is indigo with 1 square. It reaches from row 1 to row 1 and from column 1 to column 1 — 1 row by 1 column. One queen has to live inside that footprint no matter what else happens, which makes those lines the busiest real estate on the board.
Because indigo never leaves row 1, its queen must be in that row — and that instantly settles row 1 for everyone else. Sweep along it and X out every square that is not indigo. Then look at which colors just lost ground: on a grid this size that one observation usually forces a second region within a move or two.
Work outward to the other flat regions next. Mint, rose and teal are no more than two rows tall, so their queens are confined to a narrow band. When two flat regions overlap the same pair of rows, those rows are spoken for and every other color is locked out of both — the single most productive pattern in the game, and it is live on this board.
The tightest lines on puzzle #116
A quick way to find the next move is to count how many different colors touch each line. On this board, rows 1, 2, 6 and 10 are the narrowest, touched by only 3 colors. Columns 2, 3 and 12 are the tightest columns, with 3 colors.
A line with few colors is a line close to resolving. Where only 3 colors appear, one of them owns the row outright and the others must fit into the remaining 11 rows. Row 12 is the crowded extreme at 5 colors — worth revisiting later, not now.
Working through the middle game
Once two or three queens are down, the board changes character. Each queen removes an entire row, an entire column and the ring of up to eight squares around it, so the grid shrinks faster than most people expect. The habit that pays here is marking rather than remembering: after every placement, cross out the full row, the full column and all eight neighbours before hunting for the next move.
With twelve queens to seat and twelve regions competing over 144 squares, keep two questions in rotation. Does any region now have exactly one open square left? That queen is forced. Does any row or column now have exactly one open square left? That queen is forced too. Alternating between the region view and the row-and-column view is what keeps a solve moving, and it is the clearest difference between players who finish 12x12 boards quickly and players who stall halfway.
Resist the plausible-looking queen. On #116 every correct placement has a stated reason behind it, and any move you cannot justify is a guess wearing a disguise. Backing out of a wrong guess on a single-solution board takes far longer than finding the real deduction would have.
Why puzzle #116 is rated medium
The rating reflects how much the board concedes up front. Puzzle #116 has six regions confined to a single row or column, its smallest color holds 1 square, its tightest row meets just 3 colors, and the gap between its largest and smallest regions is 26 squares. Flat, lopsided boards open quickly; even, sprawling ones make you earn the first placement.
The scale has nothing to do with maths or luck — every puzzle here is solvable by deduction and has exactly one answer. It measures reasoning depth: how many eliminations must stack up before a placement becomes provable. That depth is what separates a medium board from the rest.
When a board plays tougher than its label, look for a missed mark before you look for a clever deduction. Walk back over every queen you have placed and confirm the full row, the full column and all eight surrounding squares are crossed out. The diagonal neighbours are the ones players skip, and a single skipped diagonal will hide the next forced move indefinitely.
Techniques that work well at 12x12
At 12x12 you are managing 144 squares, and holding the whole board in your head will not work. Split it into quadrants and solve locally. Most useful deductions on a grid this large are regional — a color pinched between two placed queens, or a band of rows that two regions must share. Trust the marks on the board over your memory.
Whatever the grid size, the pairing rule is the technique to internalise. When a pair of regions is boxed into the same pair of rows, they split those rows between them and nobody else may enter — and the same holds for columns, or for any n regions confined to n lines. Look for an instance of it around row 1 on this board.
A non-spoiling distribution check: solved, #116 carries 3 queens on the perimeter against 9 queens inside, 1 of them cornered. Because no two queens may touch, real solutions always look fairly evenly spread — a lopsided board in progress is a warning sign.
How to play Queens
- The board is an N×N grid divided into N colored regions.
- Place exactly one queen (♛) in each row, each column, and each colored region.
- No two queens may touch each other — not horizontally, vertically, or diagonally.
- Tap once to mark a square with an X (where a queen can't go), tap again to place a queen, and once more to clear it.
- There is always exactly one valid solution. Use logic, never guessing, to find it.
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Questions about this puzzle
How hard is 12x12 Queens puzzle #116?
Puzzle #116 is rated medium. Its smallest color region holds 1 square, its tightest row is touched by 3 colors, and six of its regions are confined to a single row or column, which gives you a forced move early.
Does 12x12 puzzle #116 have only one solution?
Yes. Every puzzle in the 12x12 collection, including #116, has been verified to have exactly one valid arrangement of twelve queens, and it can always be reached by logic alone — no guessing or backtracking needed.
Where should I place my first queen on puzzle #116?
Start with the indigo region — the most constrained color on this board, with 1 square spanning rows 1–1 and columns 1–1. Narrowing the smallest region first removes the most squares elsewhere on the grid.